Polynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation

We isolate an elementary highest-degree obstruction to polynomial changes of variables preserving unit free-Brownian diffusion. For every unital moment functional and every nonconstant noncommutative polynomial of degree \(d\), its free quadratic-variation polynomial has degree exactly \(2d-2\). The leading coefficients form a positive Gram matrix of last-letter derivatives. At any algebraically free self-adjoint tuple, this classifies self-adjoint polynomial unit-covariance maps as affine coisometries. For a standard semicircular \(m\)-tuple, the distance of the covariance from the scalars is at least \(1/\sqrt{m}\) times the squared norm of the polynomial's highest Wick component. The constant is sharp, and for quadratic polynomials this is a sharp distance-to-affine estimate. The stochastic application concerns a prescribed pathwise transformation in the same filtration; it does not resolve the more general terminal-law steering question in the AIM Free Analysis problem list.